Revise Trigonometric identities and equations for IB Mathematics: Analysis and Approaches (3.5)

An identity is an equation that holds for any value of the variable; we write it with $\equiv$. Two fundamental identities underpin almost everything in this topic, and both follow directly from the unit-circle definitions of sine and cosine.

$\text{Tangent identity:}\quad \tan\theta \equiv \frac{\sin\theta}{\cos\theta}\quad(\cos\theta\neq 0)\\ \\ \text{Pythagorean identity:}\quad \sin^2\theta+\cos^2\theta\equiv 1$

The double-angle identities relate the trig functions of $2\theta$ to those of $\theta$. They let you collapse a doubled angle into single-angle terms so that the whole equation ends up in one function. The cosine identity comes in three equivalent forms, obtained from the first by substituting the Pythagorean identity.

$\sin 2\theta = 2\sin\theta\cos\theta\\ \\ \cos 2\theta = \cos^2\theta-\sin^2\theta = 1-2\sin^2\theta = 2\cos^2\theta-1$

To PROVE an identity you start with one side — usually the more complicated one — and transform it, one justified step at a time, until it becomes the other side. You must not shuffle terms across the $\equiv$ sign as though solving an equation, because that would assume the very thing you are trying to prove. Typical moves are writing $\tan$ as $\frac{\sin}{\cos}$, applying $\sin^2\theta+\cos^2\theta=1$, and combining fractions over a common denominator.

Solving a trigonometric equation means finding EVERY angle in the stated interval that makes it true. Isolate the trig function, find the principal value with the inverse function, then use the symmetry of the unit circle and periodicity to collect all remaining solutions in range. Always check first whether the interval is in radians or degrees.

Many equations mix a squared trig term with a first-power term. Use an identity to rewrite everything in a single function; the equation then becomes a quadratic in $\sin x$ or $\cos x$, which you solve by factorising (or with the quadratic formula). Solve for the function first, then convert each value back to angles, collecting all solutions in the interval.

In Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and follow-through (FT) means a wrong value early on need not cost the marks that follow, provided your later work is correct on your own figures. Accuracy marks are dependent on the associated method mark, and the engine accepts equivalent and exact forms ($\tfrac12$, $0.5$; $\tfrac{\pi}{6}$, $30^\circ$). But this protection only exists if your method is on the page. Study how each mark below is earned by a specific line — and note that the final marks require ALL solutions in the interval.

These identities are the workhorses of the rest of trigonometry. The double-angle results reappear when you differentiate and integrate trig functions and when you simplify expressions before finding areas or volumes of revolution. The 'reduce to a single function, solve the quadratic, collect all solutions' method is exactly the routine you will reuse for equations involving $\sin 2x$, $\cos 2x$ and, at HL, the reciprocal and compound-angle identities. Master the habit — rewrite with an identity, solve the algebra, collect every solution in range, show every line — and the harder trigonometry becomes variations on a method you already own.